Finite Group Databasebeta

← #30637649

Subgroups — #30637649

Order 16256.

368 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 51 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550434C211yesmaximal, characteristic, complemented
2#22550012C211yesmaximal, characteristic, complemented
3#22550009C211yesmaximal, characteristic, complemented
4#22549786C211yesmaximal, characteristic, complemented
5#22549609C211yesmaximal, characteristic, complemented
6#22549599C211yesmaximal, characteristic, complemented
7#327×C127C211yesmaximal, characteristic, complemented
8C22×D508—21no—
9C22×D508C2211yescharacteristic, complemented
10C2×D4×D127—21no—
11C2×D1016—21no—
12C2×#14523233—21no—
13#16861194—21no—
14#16861191—21no—
15#16861183C2211yescharacteristic
16#16861181—21no—
17#16861179—21no—
18#16861177—21no—
19C2×(C2×D127)⋊C4—21no—
20C2×(C2×D127)⋊C4C2211yescharacteristic
21#16861131—21no—
22#16861129—21no—
23C22⋊(C127⋊C8)C2211yescharacteristic, complemented
24#121×C127C2211yescharacteristic
25C22⋊C8×C127C2211yescharacteristic, complemented
26#95×C127C2211yescharacteristic, complemented
27C23×D127—21no—
28C23×D127—21no—
29C23×D127D411yescharacteristic
30C2×D508—21no—
31C2×D508—21no—
32C2×D508—21no—
33C2×D508—42no—
34C2×D508—42no—
35C2×D508—42no—
36C2×D508—42no—
37C2×D508D411yescharacteristic, complemented
38C2×C22⋊D127—21no—
39C2×C22⋊D127—21no—
40C2×C22⋊D127—21no—
41D4×D127—42no—
42D4×D127—42no—
43C22×Dic127D411yescharacteristic
44C2×C4×D127—21no—
45D1016—42no—
46#14523233—42no—
47C2×C127⋊C8—21no—
48C4⋊Dic127D411yescharacteristic, complemented
49Dic127⋊C4—21no—
50C22⋊Dic127—21no—
51(C2×D127)⋊C4—21no—
52(C2×D127)⋊C4—21no—
53(C2×D127)⋊C4—21no—
54(C2×D127)⋊C4—21no—
55(C2×D127)⋊C4—42no—
56C2×D4×C127—21no—
57C2×D4×C127D411yescharacteristic, complemented
58C22×C4×C127—21no—
59C22×C4×C127C2311yescharacteristic
60C2×C8×C127—21no—
61C4⋊C4×C127D411yescharacteristic, complemented
62C22⋊C4×C127—21no—
63C22×D127—21no—
64C22×D127—21no—
65C22×D127—22no—
66C22×D127—21no—
67C22×D127—42no—
68C22×D127—41no—
69C22×D127—41no—
70C22×D127—41no—
71C22×D127—41no—
72C22×D127—41no—
73C22×D127—41no—
74C22×D127—41no—
75C22×D127—41no—
76C22×D127—81no—
77C22×D127—81no—
78C22×D127D812yescomplemented
79D508—22no—
80D508—22no—
81D508—41no—
82D508—41no—
83D508—41no—
84D508—42no—
85D508—48no—
86C22⋊D127—42no—
87C22⋊D127—41no—
88C22⋊D127—41no—
89C22⋊D127—81no—
90C2×Dic127—21no—
91C2×Dic127—21no—
92C2×Dic127—41no—
93C4×D127—42no—
94C127⋊C8—41no—
95C23×C127—21no—
96C23×C127C2×D411yescharacteristic
97D4×C127—41no—
98D4×C127—41no—
99D4×C127—41no—
100C2×C4×C127—22no—
101C2×C4×C127—21no—
102C2×C4×C127—41no—
103C2×C4×C127—41no—
104C2×C4×C127—41no—
105C2×C4×C127C2×D411yescharacteristic
106C2×C4×C127C2×D411yescharacteristic
107C8×C127—41no—
108C22×C127—21no—
109C22×C127—21no—
110C22×C127—41no—
111C22×C127—41no—
112C22×C127—41no—
113C22×C127#12011yescharacteristic
114C22×C127C2×D811yescharacteristic
115C22×C127AGL(1,ℤ/8)11yescharacteristic
116C2×D127—22no—
117C2×D127—41no—
118C2×D127—41no—
119C2×D127—42no—
120C2×D127—42no—
121C2×D127—42no—
122C2×D127—41no—
123C2×D127—41no—
124C2×D127—42no—
125C2×D127—41no—
126C2×D127—81no—
127C2×D127—84no—
128C2×D127—81no—
129C2×D127—81no—
130C2×D127—81no—
131C2×D127—81no—
132C4×C127—21no—
133C4×C127—21no—
134C4×C127—42no—
135C4×C127—41no—
136Dic127—41no—
137Dic127—41no—
138C2×C127—21no—
139C2×C127—21no—
140C2×C127—81no—
141C2×C127#44711yescharacteristic
142C2×C127#45311yescharacteristic
143C2×C127B(4,𝔽2)11yescharacteristic
144D127—42no—
145D127—82no—
146D127—81no—
147#1595—1271nomaximal, Sylow
148C127#159511yescharacteristic, complemented, Sylow
149#534—1271no—
150#447—1271no—
151#447—1271no—
152#392—1271no—
153#327—1271no—
154#327D12711yescharacteristic, complemented
155#323—1271no—
156C22×D4—1271no—
157C22×D4—2541no—
158C22×D4—2541no—
159C2×D8—2541no—
160C2×D8—2541no—
161#123—2541no—
162#123—2541no—
163#121—1271no—
164#121—2541no—
165#121C2×D12711yescharacteristic
166#120—2541no—
167#120—2541no—
168C2×C22⋊C4—1271no—
169C2×C22⋊C4—2541no—
170#102—2541no—
171#102—2541no—
172C22⋊C8—1271no—
173C22⋊C8C2×D12711yescharacteristic, complemented
174#95C2×D12711yescharacteristic, complemented
175C24—1271no—
176C24—2541no—
177C24—2541no—
178C2×D4—21no—
179C2×D4—1271no—
180C2×D4—5082no—
181C2×D4—5082no—
182C2×D4—5082no—
183C2×D4—5082no—
184C2×D4—5082no—
185C2×D4—5082no—
186C2×D4—2541no—
187C2×D4—2541no—
188C2×D4—2541no—
189C2×D4—2541no—
190C2×D4—2541no—
191C2×D4—2541no—
192C2×D4D50811yescharacteristic, complemented
193C22×C4—21no—
194C22×C4—1271no—
195C22×C4—2541no—
196C22×C4C22×D12711yescharacteristic
197D8—5082no—
198D8—5082no—
199C2×C8—21no—
200C2×C8—2541no—
201C4⋊C4—1271no—
202C4⋊C4—2541no—
203C4⋊C4D50811yescharacteristic, complemented
204C22⋊C4—21no—
205C22⋊C4—5082no—
206C22⋊C4—2541no—
207C22⋊C4—2541no—
208C22⋊C4—2541no—
209C22⋊C4—2541no—
210C22⋊C4—2541no—
211C23—21no—
212C23—1272no—
213C23—10161no—
214C23—10161no—
215C23—5081no—
216C23—5081no—
217C23—5081no—
218C23—5081no—
219C23—5081no—
220C23—5082no—
221C23—5081no—
222C23—5081no—
223C23—5081no—
224C23—2542no—
225C23—2541no—
226C23—2541no—
227C23—2541no—
228C23D4×D12711yescharacteristic
229D4—41no—
230D4—41no—
231D4—41no—
232D4—10161no—
233D4—5081no—
234D4—5081no—
235D4—5081no—
236D4—5088no—
237D4—5082no—
238D4—5082no—
239D4—5081no—
240D4—5081no—
241D4—2542no—
242D4—2542no—
243C2×C4—22no—
244C2×C4—21no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—5082no—
249C2×C4—5081no—
250C2×C4—2541no—
251C2×C4—2541no—
252C2×C4D4×D12711yescharacteristic
253C2×C4C2×D50811yescharacteristic
254C8—41no—
255C8—5081no—
256C22—21no—
257C22—21no—
258C22—41no—
259C22—41no—
260C22—41no—
261C22—10161no—
262C22—10161no—
263C22—10161no—
264C22—10164no—
265C22—10161no—
266C22—10161no—
267C22—5082no—
268C22—5082no—
269C22—5082no—
270C22—5081no—
271C22—5082no—
272C22—5081no—
273C22—5081no—
274C22—5081no—
275C22—5081no—
276C22—2542no—
277C22#1686117911yescharacteristic
278C22D8×D12711yescharacteristic
279C22#1686123611yescharacteristic
280C4—21no—
281C4—21no—
282C4—42no—
283C4—41no—
284C4—5081no—
285C4—5081no—
286C2—21no—
287C2—21no—
288C2—81no—
289C2—10162no—
290C2—10161no—
291C2—5082no—
292C2#2255001111yescharacteristic
293C2#2255004511yescharacteristic
294C2#2255006411yescharacteristic
subgroupG/H
centerC22#16861179
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#327×C127C2
socleC22×C127#120
hypercenter#327D127
hypocenterC127#1595
LCSG ▷ C22×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #327

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #22550434C2semidirect
2. #22550012C2semidirect
3. #22550009C2semidirect
4. #22549786C2semidirect
5. #22549609C2semidirect
6. #22549599C2semidirect
7. #327×C127C2semidirect
9. C22×D508C22semidirect
23. C22⋊(C127⋊C8)C22semidirect
25. C22⋊C8×C127C22semidirect
26. #95×C127C22semidirect
37. C2×D508D4semidirect
48. C4⋊Dic127D4semidirect
57. C2×D4×C127D4semidirect
61. C4⋊C4×C127D4semidirect
78. C22×D127D8semidirect
148. C127#1595semidirect
154. #327D127semidirect
173. C22⋊C8C2×D127semidirect
174. #95C2×D127semidirect
192. C2×D4D508semidirect
203. C4⋊C4D508semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#225504342
#225500122
#225500092
#225497862
#225496092
#225495992
#327×C1272
#1595127